Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem34 Structured version   Visualization version   GIF version

Theorem fourierdlem34 40778
Description: A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem34.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem34.m (𝜑𝑀 ∈ ℕ)
fourierdlem34.q (𝜑𝑄 ∈ (𝑃𝑀))
Assertion
Ref Expression
fourierdlem34 (𝜑𝑄:(0...𝑀)–1-1→ℝ)
Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)

Proof of Theorem fourierdlem34
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem34.q . . . . 5 (𝜑𝑄 ∈ (𝑃𝑀))
2 fourierdlem34.m . . . . . 6 (𝜑𝑀 ∈ ℕ)
3 fourierdlem34.p . . . . . . 7 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
43fourierdlem2 40746 . . . . . 6 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
52, 4syl 17 . . . . 5 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
61, 5mpbid 222 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 477 . . 3 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
8 elmapi 7996 . . 3 (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
97, 8syl 17 . 2 (𝜑𝑄:(0...𝑀)⟶ℝ)
10 simplr 809 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → (𝑄𝑖) = (𝑄𝑗))
119ffvelrnda 6474 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
1211ad2antrr 764 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ∈ ℝ)
139ffvelrnda 6474 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1413ad4ant14 1185 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1514adantllr 757 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
16 eleq1 2791 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑖 ∈ (0..^𝑀) ↔ 𝑘 ∈ (0..^𝑀)))
1716anbi2d 742 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑘 ∈ (0..^𝑀))))
18 fveq2 6304 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄𝑖) = (𝑄𝑘))
19 oveq1 6772 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
2019fveq2d 6308 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
2118, 20breq12d 4773 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑘) < (𝑄‘(𝑘 + 1))))
2217, 21imbi12d 333 . . . . . . . . . . . . . . 15 (𝑖 = 𝑘 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))))
236simprrd 814 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2423r19.21bi 3034 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2522, 24chvarv 2372 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2625ad4ant14 1185 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2726adantllr 757 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
28 simpllr 817 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0...𝑀))
29 simplr 809 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (0...𝑀))
30 simpr 479 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
3115, 27, 28, 29, 30monoords 39927 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) < (𝑄𝑗))
3212, 31ltned 10286 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ≠ (𝑄𝑗))
3332neneqd 2901 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
3433adantlr 753 . . . . . . . 8 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
35 simpll 807 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)))
36 elfzelz 12456 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3736zred 11595 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
3837ad3antlr 769 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
39 elfzelz 12456 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
4039zred 11595 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
4140ad4antlr 773 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
42 neqne 2904 . . . . . . . . . . . 12 𝑖 = 𝑗𝑖𝑗)
4342necomd 2951 . . . . . . . . . . 11 𝑖 = 𝑗𝑗𝑖)
4443ad2antlr 765 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
45 simpr 479 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
4638, 41, 44, 45lttri5d 39929 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
479ffvelrnda 6474 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
4847adantr 472 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
4948adantllr 757 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
50 simp-4l 825 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0...𝑀)) → 𝜑)
5150, 13sylancom 704 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
52 simp-4l 825 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0..^𝑀)) → 𝜑)
5352, 25sylancom 704 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
54 simplr 809 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0...𝑀))
55 simpllr 817 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0...𝑀))
56 simpr 479 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
5751, 53, 54, 55, 56monoords 39927 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) < (𝑄𝑖))
5849, 57gtned 10285 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑖) ≠ (𝑄𝑗))
5958neneqd 2901 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → ¬ (𝑄𝑖) = (𝑄𝑗))
6035, 46, 59syl2anc 696 . . . . . . . 8 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6134, 60pm2.61dan 867 . . . . . . 7 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6261adantlr 753 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6310, 62condan 870 . . . . 5 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) → 𝑖 = 𝑗)
6463ex 449 . . . 4 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6564ralrimiva 3068 . . 3 ((𝜑𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6665ralrimiva 3068 . 2 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
67 dff13 6627 . 2 (𝑄:(0...𝑀)–1-1→ℝ ↔ (𝑄:(0...𝑀)⟶ℝ ∧ ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗)))
689, 66, 67sylanbrc 701 1 (𝜑𝑄:(0...𝑀)–1-1→ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wne 2896  wral 3014  {crab 3018   class class class wbr 4760  cmpt 4837  wf 5997  1-1wf1 5998  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  cr 10048  0cc0 10049  1c1 10050   + caddc 10052   < clt 10187  cn 11133  ...cfz 12440  ..^cfzo 12580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581
This theorem is referenced by:  fourierdlem50  40793
  Copyright terms: Public domain W3C validator